Let $X$ and $Y$ each form a D2506: Topological metric space.

Let $d_X$ and $d_Y$ each be the D58: Metric in $X$ and $Y$, respectively.

Let $f : X \to Y$ be a D49: Proper contraction with respect to $X$ and $Y$.

Let $d_X$ and $d_Y$ each be the D58: Metric in $X$ and $Y$, respectively.

Let $f : X \to Y$ be a D49: Proper contraction with respect to $X$ and $Y$.

Then $f$ is a D55: Continuous map with respect to $X$ and $Y$.